Minimum, Maximum, First and Second
Derivatives

A tutorial on how to use calculus theorems using first and second derivatives to determine whether a function has a relative maximum or minimum or neither at a given point. These are some of the most important theorems in problem solving. This tutorial uses an interactive applet.

Theorem 1 : If function f has a relative minimum or maximum at x = a, we either have f ' (a) = 0
(stationary point) or f ' (a) does not exist.

An applet that displays the graphs of f, f ' and f '' may be used to fully understand the
theorem. Start the applet by pressing the button "press here to start".

In this applet you may expand and reflect the graph of f vertically using the button "change
height" and also shift the graph horizontally "change position". Verify that whenever there is a
relative minimum or maximum at a certain point, f '(x) is equal to zero at that point (stationary
point).